(edited for clarity)

In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 curves over a base curve X, then (to compute the cohomology of X with coefficients in the relevant local system) you have to pass to a 125-fold covering of X and compute the Jacobian of that beast."

Would someone be willing to explain this further? Given the importance of etale cohomology and the difficulties in working with it, I would really love to see the details of this spelled out.